Section 4. Objectives Determine whether the slope of a graphed line is positive, negative, 0, or undefined. Determine the slope of a line given its graph. Calculate the slope of a line given the ordered pairs of two points on the line. Page 70
SECTION 4. Slope of a Line INTRODUCTION The steepness of a mountain is sometimes measured b a grade. If ou see a sign that sas Mountain Grade 4%, it means that for ever 00 feet traveled horizontall, ou ascend or descend 4 feet. The steepness of a mountain can also be measured b the angle at which it goes up or down. A particular mountain might have an incline of 5. Think of the angle at which ou would have to tilt our head looking up from ground level to the top of a mountain. In math, we also need to measure steepness. In this section, we measure the steepness of a line. That measurement is called the slope of the line. SLOPE One of the most important properties of a line is its slope. The slope of a line is a number that describes how steep the line is. Later in this section ou will learn how to determine the number representing the slope of a line. SLOPE A number that describes the steepness (or slant) of a line. The slope of a line also describes the general direction of a line. If the slope of a line is a positive number (+), then the line looks like the uphill part of a mountain. If the slope of a line is a negative number ( ), then the line looks like the downhill part of a mountain. When we measure slope, it is important to alwas read from left to right, just like we read a sentence in English. So, alwas place our finger on the left of the line and trace the line from left to right. If the line slants up, it has a positive slope. If the line slants down, it has a negative slope. There are two other tpes of lines to consider: horizontal lines and vertical lines. The slope of a horizontal line is 0 and the slope of a vertical line is undefined. + SLOPE SLOPE Vertical Uphill Downhill UNDEFINED 0 SLOPE SLOPE Left to Right Horizontal POSITIVE SLOPE NEGATIVE SLOPE 0 SLOPE UNDEFINED SLOPE Uphill Downhill Horizontal Vertical Page 7
EXAMPLES:. Slopes of Lines that Slant Up If ou place our finger on the left of each line below and trace the line from left to right, ou notice that all the lines slant up. Therefore, all the lines have a positive slope. Slope = Slope = Slope = 3 Right now, don t worr about how we determined the numbers for the slopes. You will learn that soon.. Slopes of Lines that Slant Down If ou place our finger on the left of each line below and trace the line from left to right, ou notice that all the lines slant down. Therefore, all the lines have a negative slope. Slope = Slope = Slope = 3 PRACTICE: Determine whether the slope of each of the lines below is positive or negative... 3. ANSWERS:. positive slope. negative slope 3. positive slope Page 7
EXAMPLES:. Slopes of Horizontal Lines All horizontal lines have a slope of zero. ( slope = 0 ) Slope = 0 Slope = 0 Slope = 0. Slopes of Vertical Lines All vertical lines have an undefined slope. Undefined Slope Undefined Slope Undefined Slope PRACTICE: Determine whether the slope of each of the lines below is zero or undefined... ANSWERS:. undefined slope. slope = 0 Page 73
The chart and review video below summarize how the slope of a line relates to the direction of the line. SLOPE OF LINE DIRECTION OF LINE (from Left to Right) Positive Number Slants Up Negative Number Slants Down 0 Horizontal Undefined Vertical REVIEW: SLOPE AND DIRECTION OF A LINE PRACTICE: Determine whether the slope of each line is positive, negative, 0, or undefined.. 3. 5. 7.. 4. 6. 8. ANSWERS:. Positive 5. Negative. 0 6. Undefined 3. Negative 7. Positive 4. Undefined 8. 0 Page 74
DETERMINING SLOPE In man of the problems presented so far, the value of the slope of the line was given. For eample, Slope = 3 was written below the graph of the line. But we told ou not to worr about how that number was determined and that ou would learn the procedure later. Well, now it is time for ou to stud this topic. You will learn to determine the number that gives the slope of a line. Two methods will be shown. The first method involves determining the slope using the graph of the line, and the other method involves determining the slope without the graph. DETERMINING THE SLOPE OF A LINE FROM ITS GRAPH The slope of a line is simpl the ratio of the rise to the run. The rise is the vertical distance traveled and the run is the horizontal distance traveled as ou move from one point on the line to another point on the line. To determine the slope of an line from its graph, ou first need to identif two points on the line. An two points ou choose will result in the same answer. To find the slope, it is recommended to start at the leftmost point and then determine the rise and run needed to move to the other point. The rise is the number of units ou must move up or down, and the run is the number of units ou must move left or right. Remember that if ou move up, the rise is positive. And if ou move down, the rise is negative. The run will alwas be positive as long as ou start at the leftmost point. DETERMING SLOPE FROM A GRAPH RISE Slope = RUN = Vertical Distance (Up or Down) Horizontal Distance (Right or Left). Identif points on the line. (Points A and B). Start at the leftmost point. (Point A) 3. Determine the Rise needed to get to the other point. (Point B) Up: RISE Down: Number Number 4. Determine the Run needed to get to the other point. (Point B) RUN Right: Number Left: Number A B 5. Epress as a fraction and simplif if possible. RISE Slope = RUN Slope = Rise Run = 3 Page 75
EXAMPLE : Determine the slope of the line. B A Choose two points on the line whose coordinates are integers. Note that points A and B are not the onl two points that we could have chosen. An two points on the line will work. RISE Up A B Start at the leftmost point (Point A). Determine the rise b counting the number of units up until ou are across from the other point (Point B). We moved units. And since we moved up, the rise is positive. Therefore, Rise =. RISE Up RUN Right A B Start where ou left off at the tip of the upward arrow. Determine the run b counting the number of units to the right until ou are on the other point (Point B). We moved unit. And since we moved right, the run is positive. Therefore, Run =. Rise Slope = = = Run Epress the slope as a fraction and then simplif. What if we choose different points on the same line? Will we get the same slope value? YES! If ou choose different points on the line, ou will obtain a different fraction initiall. But once ou simplif the fraction, ou will see that the answer is the same. We show an eample like this below. RUN Right 3 RISE Up 6 A B This is the same line as Eample above. But we have chosen new points for A and B. We start at point A. We determine that the rise is 6 units up. Rise = 6 We determine that the run is 3 units to the right. Run = 3 We epress this as a fraction and simplif: Slope = Rise 6 The same answer! Run 3 Page 76
EXAMPLE : Determine the slope of the line. A Choose two points on the line whose coordinates are integers. B Note that points A and B are not the onl two points that we could have chosen. An two points on the line will work. RISE Down 4 A B Start at the leftmost point (Point A). Determine the rise b counting the number of units down until ou are across from the other point (Point B). We moved 4 units. And since we moved down, the rise is negative. Therefore, Rise = 4. RISE Down 4 A RUN Right 3 B Start where ou left off at the tip of the downward arrow. Determine the run b counting the number of units to the right until ou are on the other point (Point B). We moved 3 units right. And since we moved right, the run is positive. Therefore, Run = 3. Rise 4 4 Slope = = = Run 3 3 Epress the slope as a fraction. It cannot be simplified, but the negative sign can be moved in front of the fraction. Remember that no matter which two points ou choose on a line, ou will alwas get the same slope. Also, remember to write the slope in lowest terms b simplifing the fraction if possible. Page 77
REVIEW: DETERMINING SLOPE FROM A GRAPH Video Video PRACTICE: Determine the slope of each line. Remember that there are man pairs of points to choose on a line. For a particular line, we ma have used a different pair of points than the pair that ou select. But as long as ou simplif our fractions, ou should obtain the same answers shown in the answer section.. 4.. 5. 3. 6. Page 78
ANSWERS:. 4. Up 3 3 Slope = 3 Right Slope = 0 because the line is Horizontal. 5. Slope = Down Right Slope = Down Right 4 4 3. 6. Slope is Undefined because the line is Vertical Up Slope = Right 3 3 Page 79
DETERMINING THE SLOPE OF A LINE FROM TWO POINTS (WITHOUT A GRAPH) In the last section ou were shown the graph of a line on the coordinate plane. You learned to determine the slope of the line b identifing two points on the line and then counting the number of units needed to rise and run to move from one point to the other. But what if ou are not shown the graph of the line? Can the slope still be determined? The answer is es. In this section we will investigate how two points can be used to determine the slope of a line even if the graph is not given. The ke to the procedure is realizing the following: The rise is the number of units that we move verticall in the direction. This is the same as the difference between the -coordinates of the two points. The run is the number of units that we move horizontall in the direction. This is the same as the difference between the -coordinates of the two points. Let s illustrate with an eample. We will count the rise and run with a graph just as we did before. Then we will calculate the rise and run without a graph and show that the results are the same. WITH GRAPH RUN Right 4 We choose two points on the line. RISE Up 5 Then we determine the rise and run b counting units on the graph. We get the following: Rise Up 5 5 Slope = = = Run Right 4 4 WITHOUT GRAPH (, ) (, ) 5 7 Now, suppose that we know that the two points (, ) and (5, 7) are on a line, but we are not given the graph of the line nor a grid on which to plot the points. Rise = 7 = 5 Run = 5 = 4 Rise 5 Slope = = Run 4 To determine the rise, we calculate the difference between the -coordinates. The answer is 5, the same as the rise shown in the graph above. To determine the run, we calculate the difference between the -coordinates. The answer is 4, the same as the run shown in the graph above. We get the slope as usual using rise run. Notice that the result is the same as the slope we got above when the graph was given. Page 80
Now we will generalize the process for finding the slope of a line without a graph. As we showed, we onl need two points on the line, and an two points will work. We will call the points, and,. The small numbers in the ordered pairs are called subscripts and are used to distinguish the two ordered pairs from each other. It does not matter which ordered pair is labeled and which is labeled,,. To get the rise, we subtract the -coordinates. To get the run, we subtract the -coordinates. The slope is the ratio rise run. We epress these math operations as a formula. The formula and the procedure for finding slope are shown in the boes below. SLOPE FORMULA Rise Slope = Run Difference in -coordinates Slope = Difference in -coordinates Slope = Note: It is not necessar to see the graph to compute the slope. Onl the points (ordered pairs) are needed. CALCULATING SLOPE FROM TWO POINTS (without a graph). Label the coordinates of one point,. Label the coordinates of the other point., NOTE: It does not matter which point is labeled as and which is labeled as,,. Substitute the values of the coordinates into the slope formula: Slope = 3. Simplif the fraction if possible.. Page 8
EXAMPLES: Calculate the slope of the line that passes through each pair of points.. (, 5) and (7, 9) (, 5 ) Slope = ( 7, 9 ) 9 5 4 7 5 We label the first point and the second point,,. Substitute the values of the coordinates into the slope formula. Then simplif. What if we labeled the second point and the first point,,? Let s tr it! ( 7, 9 ) Slope = (, 5 ) 5 9 4 4 7 5 5 We reversed the wa that we labeled the points. Substitute the values of the coordinates into the slope formula. Simplif. Remember that when we divide two negative numbers, we get a positive result. The final answer is the same as above.. (4, ) and ( 5,) ( 4, ) ( 5, ), We label the first point and the second point., Slope = ( ) 54 5 4 Substitute the values of the coordinates into the slope formula. The formula has subtraction in it. In the numerator, the number following the subtraction sign is a negative number, so we must write both the subtraction sign and the negative sign. Change each subtraction to addition b adding the opposite. 3 9 Simplif to reduce the fraction to lowest terms. 3 Do not leave the negative sign in the denominator. Instead move it in front of the fraction. Page 8
3. ( 6, 3) and (,0) ( 6, 3 ) (, 0 ) Label the first point, and the second point., Slope = 0 ( 3) ( 6) Substitute the values of the coordinates into the slope formula. The slope formula has subtraction in it. Since the number following each subtraction sign is a negative number, we need to write both the subtraction sign and the negative sign. 03 6 Change each subtraction to addition b adding the opposite. 3 4 The fraction cannot be simplified. 4. (,3) and (8,3) (, 3 ) ( 8, 3 ) Label the first point, and the second point., Slope = 3 3 8 ( ) Substitute the values of the coordinates into the slope formula. The slope formula has subtraction in it. In the denominator, the number following the subtraction sign is a negative number, so we need to write both the subtraction sign and the negative sign. 33 8 Change the subtraction to addition b adding the opposite. 0 9 Simplif. 0 Remember that 0 divided b an nonzero number equals 0. Note: Whenever the -coordinates are the same, we will get 0 in the numerator, and 0 divided b an nonzero number is 0. If we were to draw the line connecting the two given points, we would get the horizontal line shown to the right. As we saw earlier, the slope of a horizontal line is 0. Page 83
5. (5,7) and (5, 4) ( 5, 7 ) Slope = ( 5, 4 ) Label the first point, and the second point 47 55 4 ( 7) 55 0 Undefined., Substitute the values of the coordinates into the slope formula. Change the subtraction to addition b adding the opposite. Simplif. Remember that an nonzero number divided b 0 is undefined. Note: Whenever the -coordinates are the same, we will get 0 in the denominator. Since division b 0 is undefined, the slope of the line is undefined. If we were to draw the line connecting the two given points, we would get the vertical line shown to the right. As we saw earlier, the slope of a vertical line is undefined. Things to Remember When Using the Slope Formula Subtracting a negative number becomes adding a positive number. Eample: ( 3) 3 0 in the numerator slope = 0 0 in the denominator undefined slope PRACTICE: Calculate the slope of the line that passes through each pair of points.. 3, and, 7., 4 and 5, 3. 8, and 8, 3 4., and 6, ANSWERS: 5. 3,5 and, 3 6. 6,6 and,4 7.,4 and,7 8. 4,9 and 5,9. 7 ( ) 5 5 3 4 4 5. 3 5 3 5 8 4 ( 3) 3. ( 4) 6 5 ( ) 6 6. 4 6 46 ( 6) 6 5 5 3. 4. 3 4 undefined slope 88 0 ( ) 0 0 6 5 7. 8. 7 4 3 undefined slope 0 9 9 0 0 5 4 9 Page 84
SECTION 4. SUMMARY Slope of a Line Slope a number that describes the steepness (or slant) of a line SLOPE Slope of Line Direction of Line (from Left to Right) Positive Number Slants Up Negative Number Slants Down 0 Horizontal Undefined Vertical RISE Slope = RUN = Vertical Distance (Up or Down) Horizontal Distance (Right or Left) DETERMINING THE SLOPE OF A LINE FROM ITS GRAPH. Identif points on the line. (Points A and B). Start at the leftmost point. (Point A) 3. Count how man units ou have to move to get to the other point (Point B): a. Count the Rise (Vertical Distance) Up: Number RISE Down: Number b. Count the Run (Horizontal Distance) Right: Number RUN Left: Number 4. Epress the slope as a fraction RISE RUN and simplif if possible. Eample: Determine the slope of the line. Rise Down 3 3 3 Slope = Run Right DETERMINING THE SLOPE OF A LINE FROM TWO POINTS. Label the coordinates of one point, and the coordinates of the other point, Note: It does not matter which point is labeled as and which is labeled as,.,.. Substitute the values of the coordinates into the slope formula: Slope =. Note: Remember to include an negative signs along with the subtraction sign. Eample: 5 ( ) 3. Simplif the fraction if possible. 0 number Note: 0 and undefined (Do the division on our calculator if ou forget these.) number 0 Eample: Find the slope of the line that passes through (3, 5) and (,9). ( 3, 5 ) (, 9 ) Slope = 9 ( 5) 9 5 4 7 3 3 4 Page 85
SECTION 4. EXERCISES Slope of a Line A B C D E. Which of the lines above have positive slope?. Which of the lines above have negative slope? 3. Which of the lines above have zero slope? 4. Which of the lines above have undefined slope? A B C D E 5. Which of the lines above have negative slope? 6. Which of the lines above have zero slope? 7. Which of the lines above have undefined slope? 8. Which of the lines above have positive slope? Page 86
Determine the slope of each graphed line. 9. 3. 0. 4.. 5.. 6. Determine the slope of the line that passes through each pair of points. 7. 3,8 and 9, 8., and,7 9.,4 and 3,4 0. 7, and 3,0. 7,4 and 7,6. 5,8 and,8 3.,6 and 5, 8 4. 7,3 and 4, 6 Page 87
Answers to Section 4. Eercises. A 3. 0. B and D 4. Undefined 3. C 5. 3 4. E 6. 0 5. C 6. D 7. A 8. B and E 7. 5 3 8. Undefined 9. 0 0. 3 9. 3. Undefined 0.. 0. 3 4 3.. Undefined 4. 9 Page 88
CHAPTER 4 ~ Linear Equations Mied Review Sections. 4.. Solve 3 8 4 0 5, graph the solution, and write the solution in interval notation. 4. Solve ab c d for b. 3. If 5 parts are defective in a shipment of 55 parts, how man parts can be epected to be defective in a shipment of 750 parts? 4. Convert 9 grams to kilograms. Use the conversion fact: kilogram (kg) = 000 grams (g) 5. What percent of 408 is 6.? 6. Sean has 4.5% deducted from each pacheck for taes. If Sean s gross salar on his last pacheck was $960, what was his net salar? 7. Write the ordered pair for each of the points shown on the graph to the right. 8. Determine if the ordered pair 6, 9. Determine the unknown coordinate so that is a solution of the equation 59., 7 is a solution of 86 0. 0. Determine the and intercepts of the graph of 4 3 4. Answers to Mied Review. (, ). c d b a 3. 50 4. 0.09 kg 5. 5% ) 6. $96.80 7. A,0 B 3, 4 C 4, 8. Yes 9. 4,7 0. -intercept: 6, 0 -intercept: 0,8 Page 89